Buckling of thin plates

Methods of Calculation of Critical Loads. In the calculation of critical values of forces applied in the middle place of a plate at which the flat form equilibrium becomes unstable and

the plate begins to buckle, the same methods as in the case of compressed bars can be used.

The critical values of the forces acting in the middle plane of a plate can be obtained by

assuming that from the beginning the plate has initial curvature or some lateral loading. Then

those values of forces in the middle plane at which deflections tend to grow indefinitely are

usually the critical values.

Another way of investigating such stability problems is to assume that plate buckles slightly

under the action of forces applied in its middle plane and then to calculate magnitudes that the

forces must have in order to keep the plate in such a slightly buckled shape. If there are no

body forces, the equation for the buckled plate then becomes

The simplest case is obtained when the forces , , are constant throughout the

plate. Assuming that there are given ratios between these forces so that = =

, and solving Eq. (1) for given boundary conditions, we shall find that the assumed buckling

of the plate is possible only for certain definite values of . The smallest of these values

determines the desired critical value.

If the forces , , are not constant, the problem becomes more involved, since Eq.

(1) has in this case variable coefficients, but the general conclusion remains the same. In such

case we can assume that the expressions for the forces , , have common factor ,

so that a gradual increase of loading is obtained by an increase of this factor. From the

investigation of Eq. (1), together with the given boundary conditions, it will be concluded that

curved forms of equilibrium are possible only for certain values of the factor and that the

smallest of these values will define the critical loading.

The energy method also can be used in investigating buckling of plates. This method is

especially useful in those cases where a rigorous solution of Eq. (1) is unknown or where we

have a plate reinforced by stiffeners and it is required to find only an approximate value of the

critical load. In applying this method we proceed as in the case of buckling bars and assume

that the plate, which is stressed by the forces acting in its middle plane, undergoes some small

lateral bending consistent with given boundary conditions. Such limited bending can be

produced without stretching of the middle plane, and we need consider only the energy of

bending and the corresponding work done by the forces acting in the middle plane of the plate.

If the work done by these forces is smaller than the strain energy of bending for every possible

shape of lateral buckling, the flat form of equilibrium of the plate is stable. If the same work

becomes larger than the energy of bending for any shape of lateral deflection, the plate is

unstable and buckling occurs. Denoting by 1 the above-mentioned work of external forces

and by U the strain energy of bending, we find the critical values of forces from the equation

Substituting for the work 1 expression

and for U expression

we obtain

Assuming that forces , , are represented by certain expressions with a common

factor , so that

A simultaneous increase of these forces is obtained by increasing . The critical value of this

factor is then obtained from Eq. (2), from which

For the calculation of it is necessary to find, in each particular case, an expression for w which

satisfies the given boundary conditions and make expression (3) a minimum, i.e., the variation

of the fraction (3) must be zero. Denoting the numerator by 1, and the denominator by 2, we

find that the variation of expression (3) is

which, when equated to zero and with 1

2= 1 gives

By calculating the indicated variations and assuming that there are no body forces, we shall

arrive at Eq. (1). Thus the energy method brings us in this way to the integration of the same

equation, which we have discussed before.

For an approximate calculation of critical loads by the energy method, we shall proceed as in

the case of truts and assume w in the form of a series

in which the functions 1(, ), 2(, ), . Satisfy the boundary conditions for w and are

chosen so as to be suitable for the representation of the buckled surface of the plate. In each

particular case we shall be guided in choosing these functions by experimental data regarding

the shape of a buckled plate. The coefficients 1, 2, . of the series must now be chosen so as

to make the expression (3) a minimum. Using this condition of minimum and proceeding as in

the derivation of Eq. (c) above, we obtain the following equations:

It can be seen from Eq. (3) that the expression for 1 and 2, after integration, will be

represented by homogeneous functions of second degree in terms of 1, 2, . . Hence Eqs. (4)

will be a system of homogeneous linear equations in 1, 2, . . Such equations may yield for

1, 2, . Solutions different from zero only if the determinant of these equations is zero.

When this determinant is put equal to zero, an equation for determining the critical value for

will be obtained. This method of calculation of critical loads will be illustrated by several

examples in the following articles.

We can arrive at Eq. (2) in another way by assuming that during buckling the edges of the plate

are prevented from movement in the xy plane. Then the displacements u and v at the

boundary vanish and the work 1 vanishes also. In such a case a lateral buckling is connected

with some stretching in the middle plane of plate, and we have to use equation

Since there are no lateral forces, 2 vanishes and we obtain

The first integral in this equation represents the charge in strain energy due to stretching of the

middle of the plate during buckling, and the second represents the energy of bending of the

plate. Eq. (5) is identical with Eq. (2), but instead of discussing the work 1, which vanishes in

this case, we have to state that the flat condition of equilibrium of the plate becomes critical

when the strain energy of stretching of the plate, released during buckling, becomes equal to

the strain energy of bending of the plate.

Buckling of Simply Supported Rectangular Plates Uniformly Compressed in One

Direction. Assume that a rectangular plate (Fig. 1) is compressed in its middle plane by forces uniformly distributed along sides x = 0 and x = a. Let the magnitude of this compressive force

per unit length of the edge be noted by . By gradually increasing we arrive at the

condition where the flat form of equilibrium of the compressed plate becomes unstable and

buckling occurs. The corresponding critical value of the compressive force can be found in this

case by integration of Eq. (1).

The deflection surface of the buckled plate can be represented, in the case of simply supported

edges, by the double series

The strain energy of bending in this case is

The work done by the compressive forces during buckling of the plate, will be

Thus Eq. (2), for determining the critical value of compressive forces, becomes

By the same reasoning as in the case of compressed bars it can be shown that expression (d)

becomes a minimum if all coefficients , except one, are taken equal to zero. Then

It is obvious that the smallest value of will be obtained by taking n equal to 1. The physical

meaning of this is that a plate buckles in such a way that there can be several half-waves in the

direction of compression but only half-wave in the perpendicular direction. Thus the expression

for the critical value of the compressive force becomes

The first factor in this expression represents the Euler load for a strip of unit width and length a.

The second factor indicates in what proportion the stability of the continuous plate is greater

than the stability of an isolated strip. The magnitude of this factor depends on the magnitude of

the ratio a/b and also on the number m, which gives the number of half-waves into which the

plate buckles. If a is smaller than b, the second term in the parenthesis of expression (e) is

always smaller than the first and the minimum value of the expression is obtained by taking

m = 1, i.e., by assuming that the plate buckles in one half-wave and that the deflection surface

has the form

The maximum deflection 11 remains indefinite plan of the plate during buckling.

The critical load, with m = 1, in expression (e), can be finally represented in the following form:

If we keep the width of the plate constant and gradually change the length a, the factor before

the parentheses in expression (g) remains constant and the factor in parentheses changes with

the change of the ratio a/b, i.e.,

For a plate of a given width the critical value of the load is smallest if the plate is square. In this

case

This is the same result obtained in considering the simultaneous action on a plate of both

bending and compression. For other proportions of the plate the expression (g) can be

represented in the form

in which k is a numerical factor, the magnitude of which depends on the ratio a/b. this factor is

presented in Fig. (2) by the curve marked m = 1. We see that it is large for small values a/b and

decreases as a/b increases, becoming a minimum for a = b and then increasing again.

Let us assume now that the plate buckles into two half-waves and that the deflection surface is

represented by the expression

We have an inflection line dividing the plate in halves, and each half is in exactly same condition

as a simply supported plate of length a/2. For calculating the critical load we can again use Eq.

(g) by substituting in it a/2 instead of a. Then

The second factor in this expression, depending on the ratio a/b, is represented in Fig. 2 by the

curve m = 2. It can be seen that the curve m = 2 is readily obtained from the curve m = 1 by

keeping the ordinates unchanged and doubling the abscissas. Proceeding further in the same

way and assuming m = 3, m = 4, and so on, we obtain the series of curves shown in Fig. 2.

Having these curves, we can easily determine the critical load and number of half-waves for any

value of the ratio a/b. It is only necessary to take the corresponding point on the axis of

abscissas and to choose the curve having the smallest ordinate for that point. In Fig. 2 the

portions of the curves defining the critical values of the load are shown by full lines. It is seen

that for very short plates the curve m = 1 gives the smallest ordinates, i.e., the smallest values

ok k in Eq. (6). Beginning with the point of intersection of the curves m = 1 and m = 2, the

second curve has the smallest coordinates; i.e., the plate buckles into two half-waves, and this

holds up to the point of intersection of the curves m = 2 and m = 3. Beginning from this point,

the plate buckles in three half-waves, and so on. The transition from m to m + 1 half-waves

evidently occurs when the two corresponding curves in Fig. 2 have equal ordinates, i.e., when

From this equation we obtain

Substituting m = 1, we obtain

At this ratio we have transition from one to two half-waves. By taking m = 2 we find that

transition from two to three half-waves occurs when

It is seen that the number of half-waves increases with the ratio a/b, and for very long plates m

is a large number. The, from (j), we obtain

i.e., a very long plate buckles in half-waves, the lengths of which approach the width of the

plate. Thus a buckled plate subdivides approximately into squares.

After the number of half-waves m in which a plate buckles has been determined from Fig. 2 or

from Eq. (j), the critical load is calculated from Eq. (g). It is only necessary to substitute in Eq. (g)

the length a/m of one half-wave, instead of a.

To simplify this calculation, Table 1 can be used; the values of the factor k in Eq. (6) are given

for various values of the ratio a/b.

From Eq. (6) the critical value of the compressive stress is

For a given ratio a/b the coefficient k is constant, and is proportional to the modulus of the

material and to square of the ratio h/b.

In the third line of Table 1 the critical stresses are given for steel plates, assuming E = 30

106 , = 0.3, and h/b = 0.01. For any other material with a modulus 1 and any other value

of the ratio h/b, the critical stress is obtained by multiplying the values in the table by the factor

It is assumed that Poissons ratio can be considered as a constant.

Comparing steel and duralumin plates of the same dimensions a and b, it is interesting to note

that for the same weight the duralumin plate will be about three times thicker than the steel

plate; since the modulus of elasticity of duralumin is about one-third that of steel, it can be

concluded from Eq. (7) that the critical stress for the duralumin plate will be about three times

larger and the critical load about nine time larger than for a steel plate of the same weight.

From this comparison it can be seen how important is the use of light aluminum alloy sheets in

such structures as airplanes where the weight of the structures is of primary importance.

The critical values of calculated by the use of Table 1, represent the true critical stresses

provided they are below the proportional limit of the material. Above this limit formula (7)

gives an exaggerated value for , and the true value of this stress can be obtained only by

taking into consideration the plastic deformation of the material. In each particular case,

assuming that formula (7) is accurate enough up to the yield point of the material, the limiting

value of the ratio h/b, up to which formula (7) can be applied, is obtained by substituting in it

= . Taking, for instance, steel for which = 40.000 , E = 30 106 , = 0.3 and

assuming that the plate is long enough so that k 4, we find from Eq. (7) that b/h 52. Below

this value of the ratio b/h the material begins to yield before the critical stress given by formula

(7) is obtained.

The edge conditions assumed in the problem discussed above are realized in the case of

uniform compression of a thin tube of square cross section (Fig. 3). When compressive stresses

become equal to their critical value (7), buckling begins and the cross sections of the tube

become curved as shown in Fig. 3b. There will be no bending moments acting between the

sides of the buckled tube along the corners, and each side is in the condition of a compressed

rectangular place with simply supported edges.

Source: Theory of Elastic Stability, Stephen P. Timoshenko, James M. Gere